Difference between social choice functions and social decision functions?

A social decision function (SDF) à la Sen (1970) is defined as a collective choice rule whose range is restricted to social preference relations which generate a choice function. From Gaertner (2009), a preference relation $$R$$ generates a choice function over a set $$X$$ if and only if $$R$$ is reflexive, complete and acyclical over $$X$$. I thus struggle to understand where exactly the difference between a SDF and a social choice function (SCF) à la Gibbard-Satterthwaite lies. A SCF is itself a choice function, so the preference relation $$R$$ generating it must satisfy the same conditions as the one generating a SDF.

• I don't think there can be a single correct answer to this question. Definitions vary across writers (and time). The onus is on the writer in a particular piece to precisely define what she means by any term.
– user18
Sep 5 '20 at 2:47

Let $$X$$ be the set of alternatives.

A social decision function maps profiles of preference orderings to relations on $$X$$ such that every nonempty subset of $$X$$ has at least one maximum under this relation.

A social choice function maps profiles of preference orderings to elements of $$X$$.

Now let $$P$$ be a profile of preferences, $$f$$ a social decision function, and $$g$$ a social choice function.

There might be more than one $$f(P)$$-maximum in $$X$$, so $$f$$, in contrast to $$g$$, will not always pin down a single choice in $$X$$.

On the other hand, suppose the alternative $$g(P)$$ is not available for some reason (say, the winning candidate died). Then $$g$$ is of no help in finding an alternative from the remaining set of alternatives $$X\setminus\{g(P)\}$$. But $$f(P)$$ will allow us to rank the alternatives in this remaining set (provided $$g(P)$$ is not the only element of $$X$$), though, again, there might be more than on $$f(P)$$-maximum in this set.

• Thank you! So in other words, the $R$ generating a social decision function is complete over all of $X$, while the one generating a social choice function must only be complete for the best alternative? What, then, is the difference between a SDF and a social welfare function? Can't we construct a social welfare function from the information encoded in a SDF (assuming acyclicality?) Sep 5 '20 at 8:59
• Edit to my comment above: is it correct that the only difference between SWFs and SDFs is that the former require transitivity ($xPy$ and $yIz \Rightarrow xPz$) while the latter only quasi-transitivity (e.g. it could be that $xPy$ and $yIz$ and $xIz$, such as in Sen's (1969) Pareto extension rule)? Sep 5 '20 at 11:55
• In Sen's 1970 book, the values of an (Arrovian) SWF have to be "orderings" (reflexive, complete, and transitive relations) on $X$, while the values of an SDF need only induce a choice function for all nonempty subsets. I don't have a characterization of that property at hand, but for infinite sets, an SWF need not be an SDF. Sep 5 '20 at 12:47